2.3 Stellar Population Synthesis Models

Stellar population synthesis models
are tools for interpreting the integrated light
(colors, line indices, and mass-to-light ratios)
that we observe from the galaxies.
Ideally, we want to determine what mix of stars give rise to
the observations.
This problem is underconstrained, however, so it is needed to make some
assumptions about how the number of different types of stars
are related.
Here we will consider so-called
single-age single-metallicity models.
In these, all the stars are formed at the same time,
with distribution in mass given by the chosen
initial mass function (IMF), and with identical chemical composition.
More advanced models take evolutionary processes into account,
e.g. enrichment of the interstellar medium, differential loss of
various element by galactic winds, time-dependent IMF, to mention a few.
However, these processes are not well understood,
and no consensus has yet been reached on these matters.

Model predictions from single-age single-metallicity models
are obtained as follows.
First it is needed to have theoretical stellar isochrones, i.e. loci in the theoretical HR-diagram
for a stellar population of a given age and
chemical composition.
Depending on the model, the chemical composition can be specified
either as just (X,Y,Z) (mass fraction of hydrogen, helium, and metals)
with the abundance ratios of the metals being solar,
or the abundances of individual metals can be taken explicitly into account.
Besides the input parameters of age and chemical composition,
to calculate an isochrone it is also necessary to have
all the physics of stellar evolution specified,
which includes opacities and how to treat convection.

Second it is needed to transform the theoretical
quantities
,
,
and
(with g being the stellar surface gravity)
to the observable quantities, i.e. colors, line indices, and
mass-to-light ratios.
This can be done on either empirical or theoretical grounds,
as will be exemplified below by the
Vazdekis et al. (1996) models and the
Tripicco & Bell (1995) models.

Third, by integrating along the isochrone weighting by the IMF
and the flux, the final values are obtained.

At this point it is warranted to define what we mean by the IMF.
Following Scalo (1986), we define the
(initial) mass spectrumf(M) as
the fraction or number of stars born per unit mass interval ,
and the
(initial) mass functionF(M) as
the fraction or number of stars born per unit logarithmic (base ten)
mass interval
.
f and F are related through
.
The logarithmic slopes of f and F evaluated at M are denoted
and ,
respectively.
They are related through
.
For power laws,
and
are independent of M.
As an example,
Salpeter (1955) found the IMF for stars in the solar neighborhood
to be reasonably well approximated by the power law
,
which has slope
.
Vazdekis et al. (1996) say that the Salpeter IMF slope is
.
Worthey (1994) says that it is x = 2.35,
but the expression he refers to is the mass spectrum in the above
language.
We will adopt the nomenclature of Vazdekis et al.